Covariance Matrix Estimation with Non Uniform and Data Dependent Missing Observations
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In this paper we study covariance estimation with missing data. We consider various types of missing data mechanisms, which may depend or not on the observed data, and have a time varying distribution. Additionally, observed variables may have arbitrary (non uniform) and dependent observation probabilities. For each missing data mechanism we use a different estimator, and we obtain bounds for the expected value of the estimation error in operator norm. Our bounds are equivalent (up to constant and logarithmic factors), to state of the art bounds for complete and uniform missing observations. Furthermore, for the more general non uniform and dependent case, the proposed bounds are new or improve upon previous results. Our bounds depend on quantities we call scaled effective rank, which generalize the effective rank to account for missing observations. We show that all the estimators studied in this work have the same asymptotic convergence rate (up to logarithmic factors).
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